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<h1>Entropy maximization</h1>
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<span class="comment">% Jo&euml;lle Skaf - 04/24/08</span>
<span class="comment">%</span>
<span class="comment">% Consider the linear inequality constrained entroy maximization problem</span>
<span class="comment">%           maximize    -sum_{i=1}^n x_i*log(x_i)</span>
<span class="comment">%           subject to  sum(x) = 1</span>
<span class="comment">%                       Fx &lt;= g</span>
<span class="comment">% where the variable is x \in \reals^{n}</span>

<span class="comment">% Input data</span>
randn(<span class="string">'state'</span>, 0);
rand(<span class="string">'state'</span>, 0);
n = 20;
m = 10;
p = 5;

tmp = rand(n,1);
A = randn(m,n);
b = A*tmp;
F = randn(p,n);
g = F*tmp + rand(p,1);

<span class="comment">% Entropy maximization</span>
cvx_begin
    variable <span class="string">x(n)</span>
    maximize <span class="string">sum(entr(x))</span>
    A*x == b
    F*x &lt;= g
cvx_end

<span class="comment">% Results</span>
display(<span class="string">'The optimal solution is:'</span> );
disp(x);
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Calling Mosek 9.1.9: 65 variables, 35 equality constraints
------------------------------------------------------------

MOSEK Version 9.1.9 (Build date: 2019-11-21 11:32:15)
Copyright (c) MOSEK ApS, Denmark. WWW: mosek.com
Platform: MACOSX/64-X86

Problem
  Name                   :                 
  Objective sense        : min             
  Type                   : CONIC (conic optimization problem)
  Constraints            : 35              
  Cones                  : 20              
  Scalar variables       : 65              
  Matrix variables       : 0               
  Integer variables      : 0               

Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries                  : 1                 time                   : 0.00            
Lin. dep.  - tries                  : 1                 time                   : 0.00            
Lin. dep.  - number                 : 0               
Presolve terminated. Time: 0.00    
Problem
  Name                   :                 
  Objective sense        : min             
  Type                   : CONIC (conic optimization problem)
  Constraints            : 35              
  Cones                  : 20              
  Scalar variables       : 65              
  Matrix variables       : 0               
  Integer variables      : 0               

Optimizer  - threads                : 8               
Optimizer  - solved problem         : the primal      
Optimizer  - Constraints            : 15
Optimizer  - Cones                  : 20
Optimizer  - Scalar variables       : 65                conic                  : 60              
Optimizer  - Semi-definite variables: 0                 scalarized             : 0               
Factor     - setup time             : 0.00              dense det. time        : 0.00            
Factor     - ML order time          : 0.00              GP order time          : 0.00            
Factor     - nonzeros before factor : 120               after factor           : 120             
Factor     - dense dim.             : 0                 flops                  : 1.08e+04        
ITE PFEAS    DFEAS    GFEAS    PRSTATUS   POBJ              DOBJ              MU       TIME  
0   1.1e+00  8.1e-01  4.3e+01  0.00e+00   1.655676798e+01   -2.581855420e+01  1.0e+00  0.00  
1   1.4e-01  9.9e-02  1.3e+00  1.02e+00   -3.813483321e+00  -8.739805594e+00  1.2e-01  0.01  
2   1.6e-02  1.2e-02  5.3e-02  1.06e+00   -5.496053237e+00  -6.066607212e+00  1.5e-02  0.01  
3   4.8e-04  3.4e-04  2.6e-04  1.01e+00   -5.697190435e+00  -5.713659083e+00  4.3e-04  0.01  
4   2.5e-05  1.8e-05  3.2e-06  1.00e+00   -5.703012589e+00  -5.703882057e+00  2.2e-05  0.01  
5   2.6e-06  1.9e-06  1.0e-07  1.00e+00   -5.703310903e+00  -5.703399767e+00  2.3e-06  0.01  
6   1.8e-07  1.3e-07  2.0e-09  1.00e+00   -5.703342595e+00  -5.703348873e+00  1.6e-07  0.01  
7   1.8e-08  1.3e-08  6.3e-11  1.00e+00   -5.703344777e+00  -5.703345393e+00  1.6e-08  0.01  
8   8.8e-09  1.8e-09  3.2e-12  1.00e+00   -5.703344991e+00  -5.703345076e+00  2.3e-09  0.01  
Optimizer terminated. Time: 0.01    


Interior-point solution summary
  Problem status  : PRIMAL_AND_DUAL_FEASIBLE
  Solution status : OPTIMAL
  Primal.  obj: -5.7033449908e+00   nrm: 7e+00    Viol.  con: 2e-08    var: 3e-09    cones: 3e-09  
  Dual.    obj: -5.7033450757e+00   nrm: 1e+00    Viol.  con: 0e+00    var: 3e-24    cones: 0e+00  
Optimizer summary
  Optimizer                 -                        time: 0.01    
    Interior-point          - iterations : 8         time: 0.01    
      Basis identification  -                        time: 0.00    
        Primal              - iterations : 0         time: 0.00    
        Dual                - iterations : 0         time: 0.00    
        Clean primal        - iterations : 0         time: 0.00    
        Clean dual          - iterations : 0         time: 0.00    
    Simplex                 -                        time: 0.00    
      Primal simplex        - iterations : 0         time: 0.00    
      Dual simplex          - iterations : 0         time: 0.00    
    Mixed integer           - relaxations: 0         time: 0.00    

------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +5.70334
 
The optimal solution is:
    0.3445
    0.3181
    0.7539
    0.8020
    0.6418
    0.3517
    0.1981
    0.2578
    0.6373
    0.3357
    0.7109
    0.8998
    0.6085
    0.6444
    0.3061
    0.4522
    0.8886
    0.7801
    0.3106
    0.6131

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